Past exam of the mathematics course of the University of Oxford / 2017 / Prelims / Mathematics III / 4 / b / ii

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- Solution ii

Solution

By the definition of a probability generating function,

G_{Y} (s) = \sum_{k \geq 1} \frac{k p _{k}}{μ} s^{k} = \frac{s}{μ} G_{Y}^{'} (s) .

Termwise differentiation of the power series is valid for

∣ s ∣ < 1

. At

s = 1

, the equality extends by the finite mean assumption, and both sides equal one.

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